If the difference of two numbers is ten and their product is eightteen what is the sum of their squares?

Answer 1

x - y = 10
xy = 18

(x - y)^2 = 10^2 = 100

but also, we have
(x - y)^2 = x^2 - 2xy + y^2
= x^2 + y^2 - 2xy
= x^2 + y^2 - 2 * 18
= x^2 + y^2 - 36

So then
x^2 + y^2 - 36 = 100
x^2 + y^2 = 136

The sum of their squares is 136.

Hope this helps.

Answer 2

x and y be the 2 numbers
x - y = 10 (let x >y)
xy = 18
find x² + y²

Expressing y in terms of x = 18/y, then substitute into
x - y =10
(18/y) - y = 10, multiply both sides by y
18 - y² = 10y
y² + 10y - 18 = 0
Using quadratic equation to solve
a=1, b=10, c=-18
y = (-10 ± 2√43)/2
y = -5 ± √43 --- here you have 2 possible answers for y
therefore,
x = 18/y = 18/(-5 ± √43) --- for each y you get an x
x² + y² = (18/(-5 ± √43))² + (-5 ± √43)²
Evaluate the equation above using either of the y

For y = -5 -√43 = -11.56
x = -1.56
x² + y² = 136

For y = -5 +√43 = 1.56
x = 11.56
x² + y² = 136 <<<< giving the same answer

Answer 3

x - y = 10
xy = 18
y = 18/x
x - 18/x = 10
x^2 - 18 = 10x
x^2 - 10x - 18 = 0
x = 5 + √43, 5 - √43
y = - 5 + √43, - 5 - √43
x^2 + y^2 = 25 + 10√43 + 43 + 25 - 10√43 + 43
x^2 + y^2 = 136